Theorem ssrdv

Description: Deduction based on subclass definition. (Contributed by NM, 15-Nov-1995)

	Ref	Expression
Hypothesis	ssrdv.1	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
Assertion	ssrdv	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ssrdv.1	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
2	1	alrimiv	⊢ ( 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
3		df-ss	⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
4	2 3	sylibr	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )